A000627 -id:A000627 - cp's OEIS frontend

A000628 Number of n-node unrooted steric quartic trees; number of n-carbon alkanes C(n)H(2n+2) taking stereoisomers into account.

1, 1, 1, 1, 2, 3, 5, 11, 24, 55, 136, 345, 900, 2412, 6563, 18127, 50699, 143255, 408429, 1173770, 3396844, 9892302, 28972080, 85289390, 252260276, 749329719, 2234695030, 6688893605, 20089296554, 60526543480, 182896187256, 554188210352, 1683557607211, 5126819371356, 15647855317080, 47862049187447, 146691564302648, 450451875783866, 1385724615285949

Offset: 0

N. J. A. Sloane

Trees are unrooted; nodes are unlabeled and have degree <= 4.
Regarding stereoisomers as different means that only the alternating group A_4 acts at each node, not the full symmetric group S_4. See A000602 for the analogous sequence when stereoisomers are not counted as different.
Has also been described as steric planted trees (paraffins) with n nodes.

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 290.
R. Davies and P. J. Freyd, C_{167}H_{336} is The Smallest Alkane with More Realizable Isomers than the Observable Universe has Particles, Journal of Chemical Education, Vol. 66, 1989, pp. 278-281.
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charles M. Blair and Henry R. Henze, The number of stereoisomeric and non-stereoisomeric paraffin hydrocarbons, J. Amer. Chem. Soc., 54 (1932), 1538-1545.
Charles M. Blair and Henry R. Henze, The number of stereoisomeric and non-stereoisomeric paraffin hydrocarbons, J. Amer. Chem. Soc., 54 (4) (1932), 1538-1545. (Annotated scanned copy)
L. Bytautats and D. J. Klein, Alkane Isomer Combinatorics: Stereostructure enumeration and graph-invariant and molecular-property distributions, J. Chem. Inf. Comput. Sci 39 (1999) 803-818, Table 1.
Lorentz Jäntschi and Lavinia-Lorena Pruteanu, Geometry of C_32 cyclic polyyne and some of its clusters, Carpathian J. Math. (2025) Vol. 41, No. 2, 371-391. See p. 371.
Pierre Leroux and Brahim Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992.
Pierre Leroux and Brahim Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 44.
R. W. Robinson, F. Harary and A. T. Balaban, The numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (1976), 355-361.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Equals A000626 + A000627.

Cf. A000598, A000602, A000625, A010372, A010373, A086194, A086200.

s[0]:=1:s[1]:=1:for n from 0 to 60 do s[n+1/3]:=0 od:for n from 0 to 60 do s[n+2/3]:=0 od:for n from 0 to 60 do s[n+1/4]:=0 od:for n from 0 to 60 do s[n+1/2]:=0 od:for n from 0 to 60 do s[n+3/4]:=0 od:s[ -1]:=0:for n from 1 to 50 do s[n+1]:=(2*n/3*s[n/3]+sum(j*s[j]*sum(s[k]*s[n-j-k],k=0..n-j),j=1..n))/n od:for n from 0 to 50 do q[n]:=sum(s[i]*s[n-i],i=0..n) od:for n from 0 to 50 do q[n-1/2]:=0 od:for n from 0 to 40 do f:=n->(3*s[n]+2*s[n/2]+q[(n-1)/2]-q[n]+2*sum(s[j]*s[n-3*j-1],j=0..n/3))/4 od:seq(f(n),n=0..38); # the formulas for s[n+1] and f(n) are from eq.(4) and (12), respectively, of the Robinson et al. paper; s[n]=A000625(n), f(n)=A000628(n); q[n] is the convolution of s[n] with itself; # Emeric Deutsch

max = 40; s[0] = s[1] = 1; s[] = 0; For[n=1, n <= max, n++, s[n+1] = (2*n/3*s[n/3] + Sum[j*s[j]*Sum[s[k]*s[n-j-k], {k, 0, n-j}], {j, 1, n}])/n]; For[n=0, n <= max, n++, q[n] = Sum[s[i]*s[n-i], {i, 0, n}]]; For[n=0, n <= max, n++, q[n-1/2]=0]; f[n] := (3*s[n] + 2*s[n/2] + q[(n-1)/2] - q[n] + 2*Sum[s[j]*s[n-3*j-1], {j, 0, n/3}])/4; Table[f[n], {n, 0, max}] (* Jean-François Alcover, Dec 29 2014, after Emeric Deutsch *)

Blair and Henze give recurrence (see the Maple code).

For even n a(n) = A086194(n) + A086200(n/2), for odd n a(n) = A086194(n).

Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20 2003

More terms from Emeric Deutsch, May 16 2004

A000626 Number of stereoisomeric paraffins with n carbon atoms.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 11, 35, 101, 290, 804, 2256, 6296, 17689, 49952, 142016, 406330, 1170272, 3390932, 9882412, 28955403, 85261439, 252213192, 749250700, 2234562027, 6688670232, 20088920730, 60525912003, 182895125096, 554186425193, 1683554604984, 5126814324715, 15647846830627, 47862034920652, 146691540312878

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric paraffin hydrocarbons, J. Amer. Chem. Soc., 54 (4) (1932), 1538-1545.
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric paraffin hydrocarbons, J. Amer. Chem. Soc., 54 (4) (1932), 1538-1545. (Annotated scanned copy)

Cf. A000627, A000628.

Reference gives recurrence.

Corrected and extended by Tom Zeller (tzellera(AT)aim.com), Jun 01 2006

cp's OEIS Frontend

A119717 Erroneous version of A000627.

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A000628 Number of n-node unrooted steric quartic trees; number of n-carbon alkanes C(n)H(2n+2) taking stereoisomers into account.

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A000626 Number of stereoisomeric paraffins with n carbon atoms.

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